Hypersurfaces in $H^n$ and the space of its horospheres

Abstract

A classical theorem, mainly due to Aleksandrov and Pogorelov, states that any Riemannian metric on $S^2$ with curvature $K>-1$ is induced on a unique convex surface in $H^3$. A similar result holds with the induced metric replaced by the third fundamental form. We show that the same phenomenon happens with yet another metric on immersed surfaces, which we call the horospherical metric. This results extends in higher dimension, the metrics obtained are then conformally flat. One can also study equivariant immersions of surfaces or the metrics obtained on the boundaries of hyperbolic 3-manifolds. Some statements which are difficult or only conjectured for the induced metric or the third fundamental form become fairly easy when one considers the horospherical metric. The results concerning the third fundamental form are obtained using a duality between $H^3$ and the de Sitter space $S^3_1$. In the same way, the results concerning the horospherical metric are proved through a duality between $H^n$ and the space of its horospheres, which is naturally endowed with a fairly rich geometrical structure.